Calculus Examples

$f (x) = 2 x - 3 x^{2}$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int 2 x - 3 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 x d x + \int - 3 x^{2} d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int x d x + \int - 3 x^{2} d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$2 (\frac{1}{2} x^{2} + C) + \int - 3 x^{2} d x$

Step 6

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$2 (\frac{1}{2} x^{2} + C) - 3 \int x^{2} d x$

Step 7

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$2 (\frac{1}{2} x^{2} + C) - 3 (\frac{1}{3} x^{3} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$2 (\frac{1}{2}) x^{2} - 3 (\frac{1}{3}) x^{3} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $2$ and $\frac{1}{2}$ .

$\frac{2}{2} x^{2} - 3 (\frac{1}{3}) x^{3} + C$

Step 8.2.2

Cancel the common factor of $2$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{2}{2} x^{2} - 3 (\frac{1}{3}) x^{3} + C$

Step 8.2.2.2

Rewrite the expression.

$1 x^{2} - 3 (\frac{1}{3}) x^{3} + C$

Step 8.2.3

Multiply $x^{2}$ by $1$ .

$x^{2} - 3 (\frac{1}{3}) x^{3} + C$

Step 8.2.4

Combine $- 3$ and $\frac{1}{3}$ .

$x^{2} + \frac{- 3}{3} x^{3} + C$

Step 8.2.5

Cancel the common factor of $- 3$ and $3$ .

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Step 8.2.5.1

Factor $3$ out of $- 3$ .

$x^{2} + \frac{3 \cdot - 1}{3} x^{3} + C$

Step 8.2.5.2

Cancel the common factors.

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Step 8.2.5.2.1

Factor $3$ out of $3$ .

$x^{2} + \frac{3 \cdot - 1}{3 (1)} x^{3} + C$

Step 8.2.5.2.2

Cancel the common factor.

$x^{2} + \frac{3 \cdot - 1}{3 \cdot 1} x^{3} + C$

Step 8.2.5.2.3

Rewrite the expression.

$x^{2} + \frac{- 1}{1} x^{3} + C$

Step 8.2.5.2.4

Divide $- 1$ by $1$ .

$x^{2} - x^{3} + C$

Step 9

The answer is the antiderivative of the function $f (x) = 2 x - 3 x^{2}$ .

$F (x) =$ $x^{2} - x^{3} + C$